For modelling the location of pyramidal cells in the human cerebral cortex, we suggest a hierarchical point process in R 3 that exhibits anisotropy in the form of cylinders extending along the z-axis. The model consists first of a generalised shot noise Cox process for the xy-coordinates, providing cylindrical clusters, and next of a Markov random field model for the z-coordinates conditioned on the xy-coordinates, providing either repulsion, aggregation or both within specified areas of interaction. Several cases of these hierarchical point processes are fitted to two pyramidal cell data sets, and of these a final model allowing for both repulsion and attraction between the points seem adequate. We discuss how the final model relates to the so-called minicolumn hypothesis in neuroscience.

This paper concerns space-sphere point processes, that is, point processes on the product space of R d (the d-dimensional Euclidean space) and S k (the kdimensional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere K-function which is a natural extension of the inhomogeneous K-function for point processes on R d to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere K-function is shown to be proportional to the product of the inhomogeneous spatial and spherical K-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere K-function is illustrated for real and simulated datasets with varying dimensions d and k.

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein and Shimatani describing a reproducing population. We discuss when closed term expressions of the first and second order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

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For modelling the location of pyramidal cells in the human cerebral cortex we suggest a hierarchical point process in R 3 . The model consists first of a generalised shot noise Cox process in the xy-plane, providing cylindrical clusters, and next of a Markov random field model on the z-axis, providing either repulsion, aggregation, or both within specified areas of interaction. Several cases of these hierarchical point processes are fitted to two pyramidal cell datasets, and of these a model allowing for both repulsion and attraction between the points seem adequate.